Gessel's equidistribution conjecture for permutations with disjoint cycle types

Let $\mu\vdash m$ and $\nu\vdash n$ be integer partitions with no common part. Let $\pi$ and $\sigma$ be permutations of cycle types $\mu$ and $\nu$, respectively, with disjoint supports. Let $A_{\pi,\sigma}$ be the subset of the conjugacy class of cycle type $\mu\sqcup\nu\vdash m+n$ consisting of the permutations for which the relative order of the letters in the union of all cycles of $\mu$ is as in $\pi$, and the relative order of the letters in the union of all cycles of $\nu$ is as in $\sigma$. Gessel's conjecture. Then

$$\sum\limits_{w\in A_{\pi,\sigma}} {\bf x}^{\Des(w)}=\sum\limits_{\tau\in \pi\shuffle \sigma} {\bf x}^{\Des(\tau)}.$$

This equidistribution identity connects descent enumerators on a constrained conjugacy class with those on shuffles of the two permutations. The conjecture was proved by Gessel, so it is included as a solved result.